It is well known that water seeks out the shortest route when traveling downhill, so why are there so many---and such large---bends in a river? Why isn't a river's path relatively straight? Hayes examines these questions, focusing on the work of Luna Leopold. Leopold thought an equation describing the direction of a river would be y = µ sin x where y is the angle from the straight down direction, µ is the maximum angle that the river makes with the straight down direction, and x is the distance along the river's centerline. Hayes gives some justification for the equation and investigates work by Hermann von Schelling that used random walks to simulate the meandering of a river. Neither these experiments nor Leopold's theory seem to give a full answer to river-meandering, however. More basic processes such as sediment transport might offer an answer, yet rivers meander even when they carry no sediment. Hayes is still intrigued by the question and although there may not be a definitive answer at the moment, he believes one does exist.

Media outlets around the world ran stories on the research described in "Better Ways to Cut a Cake", an article that appeared in the December 2006 issue of the Notices of the AMS. The article, by Steven Brams, Michael Jones, and Christian Klamer, centers on the mathematics of fair division. The "you cut, I choose" method of dividing a cake between two people has been used since time immemorial. But if the two cake eaters value the cake differently---say Alice prefers the part with chocolate icing, while Bob doesn't care which kind of icing he gets---the cut-and-choose method might leave one person less satisfied than the other. The Brams-Jones-Klamler article describes a method called "Surplus Procedure" that can result in, for example, both cake-eaters feeling that they got 65 percent of what they wanted.The research presented in the article is theoretical, but it could have applications to problems like how to divide land. Moreover, the research shows how a mathematical approach can provide better and more precise ways of resolving disputes. A three-minuteclip on the Discovery Channel illustrates the research (if the clip does not come up, do a video search on "cake").

John Hagan at Foothill High School in San Jose, CA, uses fantasy football to help his students learn algebra. The technique has gotten the class excited about algebra. Richard Salazar, one of Hagan's students, says of algebra with fantasy football, "It's not as boring as it used to be. When he connected the fantasy football it kind of had me. I want to be in math class. I wanted to do more work." Bob Creamer's eighth-grade class in Woodbine, NJ, is also using fantasy football math. Creamer is as excited as his students: In just one year, the number of eighth-grade math students judged proficient on the state exam increased from 10 percent to 54 percent.

"Not Just a Pretty Equation": Review of Mathematics and Common Sense: A Case of Creative Tension, by Philip J. Davis. Reviewed by John Ball. NewScientist, 25 November 2006, pages 54-55.

The author of this book, Philip J. Davis, is a mathematician at Brown University and a prolific commentator on many different aspects of mathematics, including how the subject is perceived by the general public. The reviewer is John Ball, a University of Oxford mathematician and president of the International Mathematical Union (IMU). Calling the book "delightful and informative", Ball uses the occasion of the review to present some compelling examples of the impact of mathematics in modern life. He also discusses the story of Grigory Perelman, the only mathematician ever to have declined the Fields Medal, commonly considered the "Nobel Prize" of mathematics. (As IMU president, Ball was the one who announced to the world, at the International Congress of Mathematicians in Madrid in August 2006, that Perelman would not accept the medal. Click here for citations of some of the worldwide media coverage of this historic event.) Ball mentions that one of the many "intelligent and approachable essays" in Davis's book discusses public perceptions of mathematics. The book recounts a story about a television reporter who once came to visit the AMS headquarters office in Providence, Rhode Island. Paraphrasing Davis's story, Ball writes: "[The reporter] had passed the building many times and often wondered what on earth went on inside. `But I really didn't want to know,' he said."

"The Biggest Questions Ever Asked" and "Journey Through Time". Special sections, NewScientist, 18 November 2006.

These special sections were published in celebration of the 50th anniversary of New Scientist magazine. The first one, "The Biggest Questions Ever Asked" does indeed tackle biggies, such as, Do we have free will?, What is Life?, and What happens after you die?. There are articles of several pages, together with short sidebars containing commentaries from an eclectic bunch of thinkers, including some mathematicians. Among the articles, two have a mathematical flavor: "What is reality?" by Roger Penrose, and "Will we ever have a theory of everything?" by Michio Kaku. Among those contributing shorter commentaries are mathematicians Marcus du Sautoy, Timothy Gowers, and Benoit Mandelbrot. The other section, "Journey Through Time", presents an overview of the past 50 years of science through a selection of New Scientist articles from this period. Judging from this collection of articles, one might conclude that nothing much happened in mathematics during those decades: There are articles on computing, theoretical physics, space exploration, genetics, chemistry, neuroscience, and nuclear power, plus several on aviation---but none exclusively on a mathematical topic.

They have been called "mountains of water" and the troughs that precede them "holes in the sea." These are a few of the phrases used to describe rogue waves, which many scientists define as waves at least 2.2 times the significant wave height---typically the average of the tallest one-third of the waves surrounding it. Some of these waves have damaged and presumably sunk large ocean-going ships. Scientists once assumed that they occurred in any one place only rarely---perhaps once every several thousand years---but "data gathered by instruments on relatively stable platforms and on buoys hint that such big waves occur much more frequently than that," writes Sid Perkins. He describes two studies in which one of the participants was Paul C. Liu, an oceanographer with the National Oceanographic and Atmospheric Administration. In one study, data that Liu and colleague Keith R. MacHutchon gathered over six years from a gas-drilling platform off the southern coast of South Africa showed that "the chance of encountering a rogue wave during any hour spent at this spot was about 3.1 percent." Four of the rogue waves were "more than four times the size of their neighbors," while one older model had suggested that one of these would occur only once every several million years!

Why do rogue waves occur? There is no single root cause. For example, Liu points to a number of likely factors for rogue waves off the coastof South Africa: a combination of the Agulhas Current flowing from the northeast, prevailing winds blowing in from the southwest, and the shape of the sea floor and coastline, all of which have an impact on wave formation. In fact, Perkins writes, scientists have"underestimated the frequency of rogue waves for many years because they presumed that real ocean waves behave as mathematically ideal waves do." For example, it was assumed that when two theoretical 1-meter-tall waves cross paths, they briefly form a wave that is 2 meters tall. In fact, this assumption is often wrong. Perkins referst o the recently published work of A. R. Osborne, an oceanographer at the University of Turin in Italy, who describes a condition called" crossing seas" in which rogue waves are more likely to form and to be taller than in a single "wave train"---a large number of waves generated by the same phenomenon that travel in a group. Later computer simulations of crossing seas done by Mattias Marklund of Umeå University in Sweden have shown the rapid generation of rogue waves. Perkins concludes that, "with such mathematical techniques, scientists may predict the conditions that spawn rogue waves." In fact, he reports, rogue wave forecasts are being issued on an experimental basis by the European Centre for Medium-Range Weather Forecasts.

Voters in California's 11th Congressional district elected Jerry McNerney, a Ph.D. mathematician and wind-energy expert, to Congress. He won the spot held by a seven-term incumbent. McNerney tells the reporter that he hopes to serve on the Energy and Commerce Committee "to put his professional background to good use," and he would also like to be on the agriculture and transportation committees. The "McNerney for Congress" website posts information about Mcnerney's background, including that he attended the University of New Mexico, where he studied engineering and mathematics, earning a Ph.D. in 1981. He was a contractor to Sandia National Laboratories, where he worked on national security programs. In 1985, he took a senior engineering position with US Windpower and in 1994 began working as an energy consultant for utility companies. McNerney will be the only mathematician in Congress. Samuel M. Rankin III, director of the AMS's Washington D.C. office, is quoted as saying: "Obviously we're very excited."

This article reports on the awarding of prizes by the Deutsche Mathematiker Vereinigung (DMV,German Mathematical Society). The renowned German literary figure Hans Magnus Enzensberger got a special prize: He had a mathematical surface named after him, the "Enzensberger Stern" (Enzensberger star), and was presented with a glass model of the six-pointed surface. Other prizewinners were George Szpiro, who received the DMV Media Prize, and Ulf von Rauchhaupt, who received the DMV Journalism Prize.

This article discusses renewed interest in twisor theory, a physical theory developed by Roger Penrose in the 1960s, which may help physicists interpret information from experiments to be conducted in 2007 at the Large Hadron Collider. Twistor theory and string theory were both candidates for a "theory of everything" that would unite physical theories about gravity and quantum mechanics. But twistor theory, Mackenzie writes, "made too many predictions that violated reality as we know it, so [the theory] crashed and burnt as a physics theory in the 1970s. Yet it lived on as a mathematical theory." The article includes a sidebar called "Six dimensions with a twist" that discusses the mathematics of twistor theory. In 2003, Edward Witten devised a way to combine string theory and twistor theory, and it is the marriage of the two that may prove relevant for next year's experiments with the Large Hadron Collider.

"The Gateway Arch is NOT a Parabola." Interview with Keith Devlin on "Weekend Edition." National PublicRadio, 4 November 2006.

Soon after a commentator on National Public Radio (NPR) identified the St. Louis Gateway arch as a parabola the station received "piles of emails" from listeners pointing out the mistake. NPR's "Math Guy" Keith Devlin was invited to explain why the arch is not a parabola but a catenary curve. Devlin starts by saying that even Galileo got the (parabola) shape wrong and that calculus (which had not yet been invented when Galileo was alive) is needed to figure it out. Devlin explains that the Gateway arch is not exactly a mathematical catenary either, because of its construction. Listen to the segment.

This time next year, a mathematician may have helped make sure your vote was counted accurately. The vote-counting scandals of the 2000 election reinvigorated the search for reliable, secure ways to tally ballots. Mathematicians and computer scientists have been working to diagnose vulnerabilities in electronic voting systems and design new voting methods using cryptography. This article provides a look at the present and future of voting systems in America, including recent research by Princeton University computer security experts on ways to load and replicate vote-changing viruses on electronic machines and ideas for ballots with encrypted portions voters can keep as receipts. Author Peter Weiss sets up the vote-counting challenge with a basic description of problems since 2000, questions about the current systems, and alternatives being developed today.

The book flows quickly and casually through advanced mathematical concepts, and its author, Martin Nowak, is a successful young Harvard professor. The reviewer did not expect to like the book but was interested and amused by Nowak's research in and explanations of evolutionary dynamics---the mathematical approach to biological phenomena like the human body's response to the AIDS virus. Portions of the book were both funny and exciting, and Nee recommends it to anyone interested in theoretical biology, though some topics and wording seemed geared toward a mathematically knowledgeable audience.

Nicolas Bourbaki was the pseudonym used by a group of mostly French mathematicians who starting in the 1930s wrote mathematics books that had a significant influence on the field. Szpiro writes that the book is written in an entertainingly and refreshingly readable style and contains not just historical but mathematical details.

If you are looking for a way to make a profit on your knowledge of conditional probability, then this short article is for you. Three cards are painted as follows: one blue on both sides, one red on both sides, and one has blue on one side and red on the other. The "mark" chooses a card. Suppose that it is blue on one side. Since this is not the red-red card, you reason with the mark that there must be a 50-50 chance that the other side of the drawn card is red. So you wager even money that the other side of the drawn card is blue. It can be shown that this is not an even bet. In fact, you will win 2/3 of the time (the blue could be from the red-blue card or from either of the two sides of the blue-blue card), and thus come out ahead by betting blue.